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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
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The Second Law of Thermodynamics states that entropy, or the amount of disorder in a system, increases each time energy is transferred or transformed. Each energy transfer results in a certain amount of energy that is lost—usually in the form of heat—that increases the disorder of the surroundings. This can also be demonstrated in a classic food web. Herbivores harvest chemical energy from plants and release heat and carbon dioxide into the environment. Carnivores harvest the...
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In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Scientists refer to the measure of randomness or disorder within a system as entropy. High entropy means high disorder and low energy. To better understand entropy, think of a student’s bedroom. If no energy or work were put into it, the room would quickly become messy. It would exist in a very disordered state, one of high entropy. Energy must be...
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Diffusion entropy analysis in billiard systems.

Gabriel I Díaz1, Matheus S Palmero1, Iberê Luiz Caldas1

  • 1Instituto de Física, IFUSP-Universidade de São Paulo, Rua do Matão, Cidade Universitária, 05314-970 São Paulo, SP, Brazil.

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Summary
This summary is machine-generated.

Shannon entropy measures diffusion exponents in dynamical systems like the standard map and oval billiard. Diffusion exponent increases when the main island area shrinks, linked to invariant tori destruction.

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Area of Science:

  • Nonlinear dynamics
  • Statistical mechanics
  • Chaos theory

Background:

  • Diffusion exponent quantifies particle spread in complex systems.
  • Shannon entropy is a measure of information or uncertainty.
  • Stickiness in phase space affects diffusion dynamics.

Purpose of the Study:

  • To investigate Shannon entropy as a tool for measuring diffusion exponents.
  • To analyze diffusion behavior in the standard map and oval billiard systems.
  • To understand the relationship between diffusion and phase space structures.

Main Methods:

  • Calculating diffusion exponents using Shannon entropy.
  • Analyzing the standard map and oval billiard phase spaces.
  • Varying nonlinear parameters to observe changes in system behavior.

Main Results:

  • Shannon entropy effectively measures diffusion exponents in the studied systems.
  • Diffusion exponent changes correlate with the area of the main island in phase space.
  • Abrupt reduction in island area leads to a significant increase in the diffusion exponent.

Conclusions:

  • Shannon entropy is a viable method for quantifying diffusion in complex systems.
  • Phase space topology, specifically island area, critically influences diffusion dynamics.
  • The destruction of invariant tori and creation of fixed points are linked to enhanced diffusion.